Find the number AB

Find the number AB

Find a number consisting of two digits: TOB which is the prime number that is in the position CDº from the list of prime numbers and that read backwards: BTO is he DCº prime number written backwards:
CD is the result of multiplying TO*B. It is the case that both TO how B how TOB They are Capicúas numbers (or palindromes) if we express them in binary. TO, B, C Y D They are all different digits.

What number is it TOB?


One way to solve the problem is by brute force obtaining the two-figure cousins ​​and ruling out cases. In this Wikipedia link, we can find a list with the first prime numbers. Since we know that both the prime number we are looking for and its position on the list are 2-digit numbers, we are limited to one of the first 26 prime numbers. We can discard 3, 5 and 7 since they are not 2-digit numbers and we can also discard all those that are below position 11 since we know that the position of the number is two digits. We can also discard the one in position 20 since the number read backwards would have only one number: 2.

We also discard those in which some of the figures in the position coincide with any of the figures in the number (for example, 31, which is in position 11) since we know that each letter:
TO, B, C Y D corresponds to a different digit. If we look at the rest of the numbers, we see that only 73 and 37 are cousins ​​with the changed digits just like 79 and 97. In the first case it is true that the position of 73 (21) has the changed digits with respect to the position of 37 ( 12) and both 3 and 7 as 73 are expressed in binary as a capicúa number (palindrome) so the number we are looking for is TOB=73.